First, I would suggest recognizing that a network is not a goal, but instead both
- a means to achieving your scientific goals and
- hopefully including a subset who you just plain enjoy interacting with
The second part can only really emerge organically, so let's focus on the first part, which is where the sort of "cold call" emails you are talking about come in. I think that you have correctly identified that "cold calls" are pretty much always ineffectual, because most people worth talking to already have an input buffer overflowing with worthless and impersonal cold calls.
If you approach the interaction from the point of view of "how can I get this person to notice me" then you're pretty much consigning yourself to the same bin. Instead, you need to have some context in which you have a legitimate reason for contacting the person that is not to be noticed. The networking and knowledge is then a natural and beneficial side effect, rather than the purpose of the interaction.
Some examples of non-in-person contexts in which such interactions commonly occur:
- Volunteering in the organization of a workshop, conference, or other academic or educational event. Even for students, there are useful roles to play, which will also end up with you encountering a bunch of researchers.
- Hosting an invited speaker: "If the mountain won't come to Muhammad then Muhammad must go to the mountain."
- Actually doing research that interacts with the other researcher's work, and either asking for their advice, or contacting them to let them know you've built off their work in a way you think they may find interesting.
I'm sure I've just scratched the surface, and there are lots of other things that can help build your network.
Finally, any of these can further be mixed with getting an introduction from a mutual acquaintance, which can help get you past the "crap-filter." If you do this, however, be aware that you are putting not just your own reputation but that of your acquaintance on the line as well.